Elementary Axioms and Pictures for Infinitesimal Calculus

is to conceive it as a system of real numbers given by decimal expansions or, more formally, as a complete ordered field. But it was not always so; prior to the formalisation of the real number concept in the late nineteenth century the number line was often considered to include infinitesimal quantities and their infinite reciprocals. It is this ambivalence which is at the heart of the new theory of nonstandard analysis and which can be exploited to give a satisfactory theory of infinitesimal calculus. Instead of one number line we must imagine two, a number system K of “constants” and a larger system K* of “quantities.” To the naked eye a pictorial representation of these would look the same, the number line of the picture above, but the number line K* contains infinitesimal detail and infinite structure not present in K.